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Relationships Between Numbers

Absolute Value

On a number line, the point representing and the point representing are both at a distance of from the point representing .
The distance between the origin and the point representing a number on the number line is called the absolute value of the number, denoted by the symbol .
For example:
  • The absolute value of is .
  • The absolute value of is also .
This is expressed as:
The absolute value of is , i.e., .

Comparing Numbers

When natural numbers are represented on a number line, numbers to the right are greater than those to the left. The same rule applies to rational numbers: numbers to the right are greater than those to the left. Hence:
  • Positive numbers are greater than negative numbers.
  • Among positive numbers, a larger absolute value corresponds to a larger number.
  • Among negative numbers, a larger absolute value corresponds to a smaller number.

Rules for Comparing Numbers

  1. Positive numbers are greater than , and negative numbers are less than . Thus, positive numbers are greater than negative numbers.
  2. Between two positive numbers, the one with the larger absolute value is greater.
  3. Between two negative numbers, the one with the larger absolute value is smaller.
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